Roger W. Anderson

Inversion of orbiting scattering from elastic collisions of reactive molecules

For elastic scattering produced by an attractive, spherically symmetric potential, the classical deflection angle is a single‐valued function for impact parameters equal to or greater than that which gives orbiting at the centrifugal barrier. If reaction removes contributions from smaller impact parameters, the potential in this outer region can be derived from the elastic scattering by a simple inversion procedure. This is obtained by slightly modifying the Firsov method, so as to integrate from small angles inwards. Close to the centrifugal barrier, the scattering is determined primarily by the first and second derivatives of the potential, and an approximate inversion method for obtaining the derivatives is also given. These procedures are applied to data for nonreactive scattering of K atoms from Cl2, Br2, I2, and ICl. The results indicate the orbiting model is a useful approximation for such reactions in the thermal range, although a less simple treatment is necessary at higher energies. The potentia...

The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors.

This work demonstrates that the quantum mechanical moments of a state described by the density matrix correspond to discrete spherical harmonic moments of the classical multipole expansion of the spatial distribution of the angular momentum vectors. For the diagonal density matrix elements, this work exploits the fact that the quantum mechanical vector coupling (Clebsch-Gordan) coefficients become increasingly accurate discrete representations of spherical harmonics as j increases. A Schwinger-type basis accounts for nonaxially symmetric angular distributions, which result in nonzero off-diagonal elements of the density matrix. The resulting discrete minimum uncertainty picture of the classical moments has a stringent equivalence with the quantum mechanical one for all j and provides an unambiguous connection for the classical and quantum moments in the large j limit. The equivalence is numerically tested for simple models, and there is a satisfying equivalence even for small j. Applications, implications, and extensions are indicated, and the relevance of this work for the interpretation of classical mechanical simulations of inelastic and reactive molecular collisions will be documented elsewhere.

3nj Morphogenesis and semiclassical disentangling.

Recoupling coefficients (3nj symbols) are unitary transformations between binary coupled eigenstates of N = (n + 1) mutually commuting SU(2) angular momentum operators. They have been used in a variety of applications in spectroscopy, quantum chemistry and nuclear physics and quite recently also in quantum gravity and quantum computing. These coefficients, naturally associated to cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and analytical features that make them fascinating objects to be studied on their own. In this paper we develop a bottom-up, systematic procedure for the generation of 3nj from 3(n - 1)j diagrams by resorting to diagrammatical and algebraic methods. We provide also a novel approach to the problem of classifying various regimes of semiclassical expansions of 3nj coefficients (asymptotic disentangling of 3nj diagrams) for n > or = 3 by means of combinatorial, analytical and numerical tools.

Exact computation and large angular momentum asymptotics of 3nj symbols: Semiclassical disentangling of spin networks

Spin networks, namely, the 3nj symbols of quantum angular momentum theory and their generalizations to groups other than SU(2) and to quantum groups, permeate many areas of pure and applied science. The issues of their computation and characterization for large values of their entries are a challenge for diverse fields, such as spectroscopy and quantum chemistry, molecular and condensed matter physics, quantum computing, and the geometry of space time. Here we record progress both in their efficient calculation and in the study of the large j asymptotics. For the 9j symbol, a prototypical entangled network, we present and extensively check numerically formulas that illustrate the passage to the semiclassical limit, manifesting both the occurrence of disentangling and the discrete-continuum transition.

Crossed molecular beam‐tunable laser determination of velocity dependence of intramultiplet mixing: K(4p2P1/2)+He →K(4p2P3/2)+He

The velocity dependence of intramultiplet mixing, K(4p2P1/2) +He→K(4p2P3/2)+He, has been measured over the relative velocity range v=1.3–3.4 km/sec. The cross section appears to fit a linear function Q (v) =A (v−v0), where a=6.3×10−4 A2 and v0= 7.9×104 cm/sec. The value of A is obtained by normalization to the literature thermal average cross section. The intramultiplet mixing theory of Nikitin is modified to yield Q (v) for the process. The modified theory correctly exhibits detailed balancing, and it is normalized to provide a very good fit to the observed Q (v). The magnitude of the normalization factor, however, is larger than that predicted from recent pseudopotential calculations of the excited state potentials. The temperature dependence of intramultiplet mixing is predicted. The use of laser polarization to determine the mj dependence of the process K(4p2P3/2+He→K(4p2P1/2)+He and other collision processes of excited 2P3/2 states is examined.

Solution of vibrational excitation problems with constant step size Magnus propagators: Convergence, perturbation analysis, and approximate decoupling

Constant step size Magnus propagators are used to integrate the Secrest–Johnson vibrational excitation problem with parameters given by Stechel, Walker, and Light. Calculations are done for a six channel basis for E=6 and 8 and for a 30 channel basis for E=60. For all the calculations the error scales as the fourth power of the step size for small steps. This indicates a significant cancellation between the errors in the Magnus propagation and the assumed diagonalization of the potential over each interval. The error grows rapidly when the step size is such that at least one channel is propagated by near half a wavelength. The errors in the E=6 calculation are compared with those for the log derivative, renormalized Numerov, R matrix propagation methods, and a quadratic approximate potential method. The constant step size Magnus method is superior to the other methods for this vibrational excitation problem. A perturbation analysis is presented to show why accurate calculations are possible with large non...

Exact and asymptotic computations of elementary spin networks: classification of the quantum---classical boundaries

Increasing interest is being dedicated in the last few years to the issues of exact computations and asymptotics of spin networks. The large---entries regimes (semiclassical limits) occur in many areas of physics and chemistry, and in particular in discretization algorithms of applied quantum mechanics. Here we extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient, enlightening the insight gained by exploiting its self---dual properties and studying it as a function of two (discrete) variables. This arises from its original definition as an (orthogonal) angular momentum recoupling matrix element. Progress also derives from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational advances ---based on traditional and new recurrence relations--- which allow an interpretation of the observed behaviors in terms of an underlying Hamiltonian formulation as well. This paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines (caustics) in the plane of the two matrix labels. This analysis marks the evolution of the turning points of relevance for the semiclassical regimes and puts on stage an unexpected key role of the Regge symmetries of the 6j.

Representation in hyperspherical and related coordinates of the potential-energy surface for triatomic reactions

Useful formulae for the treatment in hyperspherical and related coordinates of the elementary chemical reaction as a three-body problem are reported. In the kinetic plane (where one-dimensional potential-energy surfaces are represented) besides rotations describing all rearrangements, also those pertaining to the heliocentric Radau–Smith representation are exhibited. Explicit relations are given between physical quantities (distances, bond angles) and the two basic hyperspherical parametrisations. They are based on a formulation of internal coordinates which leads also to the systematic study of alternative ways for the graphical representation of three-dimensional potential-energy surfaces. The usefulness and some properties of kinetic paths (essentially two-dimensional plots of specific features of the potential-energy surface as a function of the hyperradius) are finally presented. Alternative notations employed in the recent literature on chemical kinetics are compared in an Appendix.

Quantum calculation of intramultiplet mixing: Comparison with simple radial and angular mechanisms

Cross sections for intramultiplet mixing in the lowest 2P states of alkali atoms (K, Rb, Cs) in collisions with He atoms are calculated in body fixed coordinates. The cross sections for transitions that maintain or change the magnitude of the projection of the electronic angular momentum are compared with very simple semiclassical models for radial (R1) and angular (R2) coupling of electronic states. The angular mixing model uses a hard sphere interaction. The agreement is good for the total cross sections and for the transition probabilities for different partial waves. The total cross sections are compared with experimental and other theoretical results, and cross sections are presented for collisions of polarized 2P3/2 atoms. The quantum mechanical equations are integrated with a constant step size Magnus method, and we find uniform convergence O(h2) for small step sizes. The error rapidly increases for steps large enough to allow propagation of a channel by half of a de Broglie wavelength.

Longitudinal Plasma Resonance Shifts in Gold Nanoparticle Aggregates

Recent interest in colloidal gold focuses on understanding the tunability of the longitudinal and transverse plasma resonance. It was reported that the reduction of HAuCl4 by Na2S produces gold nanoparticles with an optical absorption in the near infrared. This absorption blue shifts during the course of the reaction. X-ray photoemission spectroscopy (XPS) measurements on this system indicated that there was little sulfur present in the system. A small angle x-ray scattering (SAX) experiment was used to monitor the reaction while simultaneously the UV-VIS spectrum was measured. During the reaction the fractal dimension decreased from 4.154 ± 0.850 to 0.624 ± 0.146. The decrease in fractal dimension coincided with the blue shift in the longitudinal plasma resonance from the near IR to the visible. This suggests a change from reaction limited colloid aggregation (RLCA) to diffusion limited colloid aggregation (DLCA), caused the shift in the plasma resonance.